FOUR elements crystallize in the diamond cubic
structures: silicon, germanium,α-tin and diamond itself.
It can be thought of as an f.c.c. structure with two atoms associated with each
f.c.c. atom position. Each atom has four nearest neighbours to which it is
linked by four purely covalent bonds. The low co-ordination makes it a
very open structure, with a low density. But despite this, the covalent
elements are enormously strong, having a flow strength at which exceedsμ/20 at 0 K and which remains high even at 0.5 TM. Their mechanical properties are interesting because
they typify the extreme behaviour associated with pure covalent bonding.

Maps
and data for silicon and germanium are shown in Figs. 9.1 to 9.4. The parameters
used to compute the maps are listed in Table 9.1.

TABLE
9.1 The covalent elements

Material

Silicon

Germanium

Crystallographic
and thermal data

Atomic
volume, Ω (m3)

Burger's
vector, b (m)

Melting
temperature, TM(K)

2.00 x 10-
29

3.83 x 10-
10

1687

2.26 x 10-
29

3.99 x 10-
10

1211

Modulus

Shear
modulus at 300 K μ0 (MN/m2)

Temperature
dependence of modulus,

6.37 x 104

-0.078

(a)

(b)

5.20 x 104

-0.146

(g)

(b)

Lattice
diffusion (normal)

Pre-
exponential, D0υ(m2/s)

Activation
energy, Qυ(kJ/mole)

0.9

496

(c)

7.8 x 10-
4

287

(h)

Boundary
diffusion

Pre-
exponential, δD0b
(m3/s)

Activation
energy, Qb(kJ/mole)

10-
l5

300

(d)

10-
17

172

(d)

Core
diffusion

Pre-
exponential, acD0c (m4/s)

Activation
energy, Qc(kJ/mole)

10-
25

300

(d)

10-
27

172

(d)

Power-
law creep

Exponent, n

Dorn
constant, A

5.0

2.5
x 106

(e)

5.0

1.0 x 108

Obstacle-controlled
glide

0
K flow stress,

Pre-exponential, (s-1)

Activation
energy, ∆F/μ0b3

—

—

—

—

—

—

Lattice-resistance-controlled
glide

0
K flow stress,

Pre-exponential, (s-1)

Activation
energy,∆F/μ0b3

0.07

1011

0.2

(f)

0.06

1011

0.2

(f)

(a) McSkimin
(1953); Prasad and Wooster (1955);

(b) McSkimin
(1959); McSkimin and Andreatch (1964);

(c) Masters
and Fairfield (1966);.

(d) Estimated
using Qb=0.6 Qυ and
δD0b=2bD0υ; δDb=

δD0b exp –(Qb/RT);
acDc=acD0c exp - (Qc/RT).

(e) Adjusted
to fit Myschlyaev et al. (1969). The constant Arefers
to tensile creep; in computing the maps A is used.

(f) Adjusted
to fit the hardness data of Trefilov and Mil'man (1964).

(g) Fine
(1953); McSkimin and Andreatch (1963).

(h) Letaw et
al. (1956).

(i) Estimated
by using the same creep exponent as silicon and
fitting the constant A to the data of Gerk (1975); see note
(e).

The mechanical behaviour of Si and Ge is
dominated by their localized, strongly directional, covalent bonding. It
creates an exceptionally large lattice resistance for slip on all slip systems,
includ­ing the primary system (Alexander and Haasen, 1968). The
result is that the normalized flow strength at below half the melting point is
larger than that for any other class of solid (except perhaps ice). It is so
large that, in most modes of loading, covalent elements and compounds fracture
before they flow. The hardness test allows the plastic behaviour to be studied
in a simple way down to 0.2 TM,when fracture becomes a problem here also.

The localized nature of the bonding has
other consequences: the energy of formation and motion of point defects is
large, so that diffusion (at a given homologous temperature) is slower than in
other classes of solid. At high temperature ( > 0.6 TM)Si
and Ge show power- law creep, often complicated by an inverse transient
(in creep tests) or a yield drop (in constant strain- rate tests) during
which the dislocation density increases towards a steady state. This steady
state is rarely reached in the small strains of most published experiments,
although Myshlyaev et al. (1969) [1] report steady
behaviour in silicon. We have assumed that steady-state creep can be
described by the rate-equation used in earlier sections for metals (eqn.
(2.21)) and that, at suffici­ently low stresses, diffusional flow (as described
by eqn. (2.29)) will appear . Both creep mechanisms are diffusion-controlled,
and are slow because of the slow rates of diffusion. Thus, the covalent
elements and the covalently bonded 3-5 and 2-6 compounds, as a
class, are stronger (in terms of the normalized variable σs/μ) at all homologous temperatures, than any other class of solid; the
ultimate example, of course, is diamond. The maps are a tolerably good fit to
the hardness data except below 0.2 TM(when fracture often occurs round the indenter) and to
the high- temperature (>0.8 TM) creep data. At inter­mediate temperatures, the maps
match the creep data less well. Lower yield point and transient creep data,
some of which are shown on Fig. 9.3, deviate widely from the predicted steady-state behaviour of the maps (as expected) and were ignored in deriving the parameters
used in their construction.

The stress axes of the maps shown in this
chapter have been extended upwards by a factor of 10 compared with the others
in this book. This is because, for the reasons already given, the data lie at
higher normalized stresses than those for most other materials; and because it
allows us to show where certain pressure- induced phase transformations
would occur in simple compression (using the pressure that would appear in a
uniaxial compression test). These phase trans­formations could become
important in hardness tests, when the hydrostatic pressure at yield is roughly
three times larger for the same equivalent shear stress, σs.

There is extensive literature on the
mechanical properties of silicon and germanium. The interested reader should
refer to Alexander and Haasen (1968) [2] for a
comprehensive review.

Dislocations and slip systems in diamond
cubic crystals have been reviewed by Alexander and Haasen (1968) [2]. Under
normal circumstances, slip occurs on , giving the five independent systems
necessary for polycrystal plasticity.

The low-temperature yield strength (T
< 0.5 TM) is based on the micro- hardness measurements of
Trefilov and Mil'man (1964) [7] corrected
according to Marsh (1964) [8]. In both
cases the temperature dependence of the hardness is less than can be explained
using realistic values of ∆Fp, and the rate­equation, using the constants of Table
9.1, does not match the measured hardness below 0.2 TM very
well (this may be because fracture occurred at the indenter).

The power- law creep constants for
silicon were fitted to the steady- state single- crystal creep data
of Myshlyaev et al (1969) [1]. In the
range of their measurements they observed an activation energy of between 400
and 520 kJ/mole; that for self­diffusion is 496 kJ/mole (Masters and
Fairfield, 1966) [9] which
therefore provides an approximate description of their results. The use of a
power- law, too, is an approximation (Fig. 9.2); our values of n =
5 and A = 2.5 x 106 provide the best fit.

We have found no steady- state creep
data for germanium. Brown et al. (1971) [10], studying
transient creep, found an activation energy of around 340 kJ/mole, which is
rather larger than that for lattice diffusion of 287 kJ/mole (Letaw et al.,
1956) [11]. Useful
information is contained in the high- temperature hardness measurements of
Gerk (1975) [12] and
Trefilov and Mil'man (1964) [7] which are
plotted on Figs. 9.3 and 9.4. Gerk's data suggest a power- law with n
= 3; but because the hot- hardness test does not approach steady-state
conditions, we have fitted the data to a power- law with the same
exponent, n, as that for silicon (namely 5), adjusting the constant A to fit Gerk's data. This choice leads to better agreement with the hardness data of Trefilov and Mil'man (1964) [7] than the
alternative choice of n = 3 suggested by Gerk (1975) [12].